infinitely-differentiable function that is not analytic

If f∈𝒞∞, then we can certainly write a Taylor series for f. However, analyticity requires that this Taylor series actually converge (at least across some radius of convergence) to f. It is not necessary that the power series for f converge to f, as the following example shows.

Let

f⁢(x)={e-1x2x≠00x=0.

Then f∈𝒞∞, and for any n≥0, f(n)⁢(0)=0 (see below). So the Taylor series for f around 0 is 0; since f⁢(x)>0 for all x≠0, clearly it does not converge to f.